Non-empty intersection of longest paths in $H$-free graphs

TL;DR

This work advances the understanding of which graphs $H$ yield a Gallai family for $\mathrm{Free}(H)$ by showing a necessary condition that $H$ be a linear forest with $|V(H)| \le 9$, and proving sufficiency for all 4-vertex linear forests. It establishes a sequence of fixer results, proving that $P_3+P_1$, $P_2+2P_1$, $4P_1$, and $5P_1$ are fixers, and develops a general CE-type theorem stating that large $k$-connected graphs with $\alpha(G) \le k+2$ have all maximum-degree vertices Gallai. The paper also situates these results within a broader program to classify monogenic Gallai families, clarifies the structure of potential $9$-vertex fixers, and identifies several open problems and conjectures, including optimal bounds and the precise set of fixers among linear forests induced by a key counterexample $G_0$.

Abstract

We make progress toward a characterization of the graphs $H$ such that every connected $H$-free graph has a longest path transversal of size $1$. In particular, we show that the graphs $H$ on at most $4$ vertices satisfying this property are exactly the linear forests. We also show that if the order of a connected graph $G$ is large relative to its connectivity $κ(G)$, and its independence number $α(G)$ satisfies $α(G) \le κ(G) + 2$, then each vertex of maximum degree forms a longest path transversal of size $1$.

Figures (6)

• Figure 1: The graph $G_0$: A $12$-vertex graph with no Gallai vertex.
• Figure 2: The linear forests on $4$ vertices. These are exactly the graphs $H$ on $4$ vertices such that $\mathrm{Free}(H)$ is a Gallai family.
• Figure 3: Construction of $A$ in the proof of \ref{['lem:indep-set']}.
• Figure 4: Part 3 in \ref{['lem:minihammer']}.
• Figure 5: Case $k = 2$ in the proof of \ref{['lem:special-to-Gallai']}.

Theorems & Definitions (38)

• Proposition 1
• proof
• Remark 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Theorem 6